The Celsius and Fahrenheit Scales — Question 2

Question 2

A difference channel must not inherit the scale offset

Use numerical readings $F=\frac95C+32$ and $T/\mathrm K=C+273.15$; these conversions are exact. A differential instrument measures a bath above its reference by $\Delta C=12.0$. Firmware mistakenly reports the difference as $\Delta F=1.8\Delta C+32$. The reference is at $20.0^\circ\mathrm C$.

1. Find the correct difference in Fahrenheit degrees and the actual bath readings on all three scales.

2. If the incorrect difference is added to the correct Fahrenheit reference reading, find the resulting temperature error in Celsius degrees.

Question 2 - Solution

A difference channel must not inherit the scale offset

(a) Method and calculation

$$\Delta F=F_b-F_r=1.8(C_b-C_r)=1.8(12)=21.6.$$$$C_b=32.0,\quad F_b=89.6,\quad T_b=305.15\,\mathrm K.$$

(b) Interpretation and further calculation

The reference is $68.0^\circ$F, while the false difference is $53.6$ Fahrenheit degrees.$$F_{\mathrm{false}}=68+53.6=121.6,\qquad C_{\mathrm{false}}=(121.6-32)/1.8=49.7778.$$$$C_{\mathrm{false}}-C_b=17.7778=32/1.8.$$The reference offset has been counted twice.

Physical check

Offsets cancel in differences and derivatives, but remain in absolute readings.